Thermoelectric Ratchet Effect for Charge Carriers with Hopping Dynamics

We show that the huge Seebeck coefficients observed recently for ionic conductors arise from a ratchet effect where activated jumps between neighbor sites are rectified by a temperature gradient, thus driving mobile ions toward the cold. For complex systems with mobile molecules like water or polyethylene glycol, there is an even more efficient diffusiophoretic transport mechanism, proportional to the thermally induced concentration gradient of the molecular component. Without free parameters, our model describes experiments on the ionic liquid EMIM-TFSI and hydrated NaPSS, and it qualitatively accounts for polymer electrolyte membranes with Seebeck coefficients of hundreds of $k_B/e$.

Figure 1

(a) Gibbs enthalpy landscape of a mobile charge in a conduction channel. The carrier passes from one site to the other by thermally activated jumps over the barrier. (b) Nonequilibrium hopping dynamics between neighbor sites. The temperature in the left well is higher than in the right one, $T_1>T_2$, resulting in stronger thermal fluctuations and more frequent jumps to the right, ${\mathrm{\Gamma }}_{12}>{\mathrm{\Gamma }}_{21}$. (c) Gibbs enthalpy landscape of an isolated double-well potential.

Figure 2

(a) Thermodiffusion in a connected network like in Fig. 1, for example of sodium ions in NaPSS [25]. The cations diffuse toward the cold boundary and leave uncompensated ${\mathrm{PSS}}^{-}$ at the hot side, thus inducing a thermoelectric field $E=S\nabla T$. (b) In ionic liquids, charge transport relies on vacancy formation [30]; initial and final states are shown schematically in Fig. 1. We show the case realized in ${\mathrm{EMIM}}^{+}\text{−}{\mathrm{TFSI}}^{-}$, where anions (yellow) accumulate at the cold side, and cations (green) at the hot.

Figure 3

Composition dependence of the activation enthalpy of mobile ions. Red circles show data of De et al. [25] for sodium ions in the polyelectrolyte complex ${\mathrm{NaPSS}}_{0.55}\text{−}{\mathrm{PDADMAC}}_{0.45}$ with water weight fraction $c$. Black circles and squares give data for ${\mathrm{EMIM}}^{+}$ and ${\mathrm{acetate}}^{-}$, with or without added glucose, as measured by D’Agostino et al. [34]. The solid lines indicate linear fits, with slope $H_c=-2.75\mathrm{eV}$ (Na), 1.35 eV (EMIM), and 1.66 eV (acetate).